Positron emission tomography (PET) is a branch of nuclear medicine in which a positron-emitting radiopharmaceutical is introduced into the body of a patient. As the radiopharmaceutical decays, positrons are generated. More specifically, each of a plurality of positrons reacts with an electron in what is known as a positron annihilation event, thereby generating a coincident pair of gamma photons which travel substantially in opposite directions along a line of coincidence. A gamma photon pair detected within a coincidence time is ordinarily recorded by the PET scanner as an annihilation event.
In time-of-flight (“TOF”) imaging, the time within the coincidence interval at which each gamma photon in the coincident pair is detected is also measured. The time of flight information provides an indication of the location of the detected event along the line of coincidence. Data from a plurality of annihilation events is used to reconstruct or create images of the patient or object scanned, typically by using statistical (iterative) or analytical reconstruction algorithms.
FIG. 1 illustrates the transaxial and axial coordinates of an emitted positron and the measured line of response (LOR) of a 3D detector. The coordinates (xe, ye, ze) or (se, te, ze) define the emitted positron's image coordinate. The measured LOR's projection coordinate can be defined by either (s, φ, z, θ), where z=(za+zb)/2, or may include the additional dimension t for a TOF-LOR.
In PET, random coincidences occur due to the finite width of the coincidence window, which is used to detect true coincidences. If two uncorrelated single events are detected within the coincidence window, they can mistakenly be identified as a true coincidence event and recorded. The rate of random events is proportional to the single event rate on each detector and the size of coincidence window, as shown in Equation (1):Cij=2τriτj  (1)in which Cij represents the random coincidences count rate on the LOR that connects the i-th and j-th detectors, ri, rj represents the single count event rates of the i-th and j-th detectors, and τ is the coincidence window size.
Random coincidences can comprise a large portion of the recorded prompt coincidence events (which include true, scatter, and random coincidence events), especially in the operation of a 3D PET scanner and high radioactivity concentration. If not compensated for properly, random coincidence events can introduce substantial quantitative errors in the reconstructed images.
The most accurate and commonly used method for random estimation involves the use of a delayed coincidence window. The delayed coincidence window can remove the correlation of two single events in each recorded pair event. Since the delayed coincidence window is usually postponed several ten-to-hundred times the coincidence window size, the possibility of two recorded single events being from a single annihilation is very rare. Therefore, in this method, only random events are recorded.
Recently, a time-of-flight mask was designed to reduce random events in the prompt data by completely filtering out those random coincidences that provide no contribution to the reconstruction field-of-view (FOV). In this process, random events with a TOF difference out of the mask will be removed before reconstruction starts, which leads to more accurate reconstructed images in list-mode reconstruction and lower computation time.
In TOF list-mode reconstruction, the random events with TOF information are estimated in the following way. The non-TOF delay list-mode data are first rebinned and interpolated to a 4D interp-sinogram, smoothed, and then back-interpolated to a 4D raw-sinogram. In the last step, a 5D TOF raw-sinogram is generated by spreading the LOR counts evenly into tangential t-bins, and considering the difference in t-ranges along the LOR's radial direction and oblique angle.
However, with the TOF mask applied, the random distribution will be changed with the introduced circular-edged TOF mask, which causes more random events collected in the central regions of field-of-view (FOV) and fewer random events collected at the edges of FOV. The previously described procedure without TOF mask is not applicable in this case. The reasons are (1) the random smoothing step will change the random distribution over the s dimension, which may bring in estimation error during the back-interpolation step, and (2) when spreading the LOR counts evenly into each tangential t-bin, it is assumed that the random events counts are uniformly distributed along the t dimension, which is not true after applying the TOF mask. Thus, the random events are collected more in the center of the FOV, but fewer are collected at the circular edges of the TOF mask.